LECTURE 1 Introduction When a group G acts on a manifold M, one would like to understand the relation between: • algebraic properties of the group G, • the topology of M, • the G-invariant geometric structures on M, and • dynamical properties of the action (such as dense orbits, invariant mea- sures, etc.). If we assume that G is a connected Lie group, then the structure theory (A1.3) tells us there are two main cases to consider: • solvable Solvable groups are usually studied by starting with Rn and pro- ceeding by induction. • semisimple There is a classification that provides a list of the semisimple groups (SL(n, R), SO(p, q), etc.), so a case-by-case analysis is possible. Alternatively, the groups can sometimes be treated via other categorizations, e.g., by real rank. The emphasis in these lectures is on the semisimple case. (1.1) Assumption. In this lecture, G always denotes a connected, noncompact, semisimple Lie group. 1A. Discrete versions of G A connected Lie group may have discrete subgroups that approximate it. There are two notions of this that play very important roles in these lectures: • lattice This is a discrete subgroup Γ of G such that G/Γ is compact (or, more generally, such that G/Γ has finite volume). • arithmetic subgroup Suppose G is a (closed) subgroup of GL(n, R), and let Γ = G ∩ GL(n, Z) be the set of “integer points” of G. If Γ is Zariski dense in G (or, equivalently, if G ∩ GL(n, Q) is dense in G), then we say Γ is an arithmetic subgroup of G. Discrete versions inherit important algebraic properties of G: 1 http://dx.doi.org/10.1090/cbms/109/01

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